<header>
    基本概念与性质
</header>
<p>
    <span class="title">
        定义
    </span>
    设{a<sub>n</sub>}为数列，a为定数。若对任给的正数ε，总存在正整数N，使得当n＞N时有
    <span class="oneline">
        |a<sub>n</sub> - a| ＜ ε
    </span>
    则称为数列{a<sub>n</sub>}收敛于a，定数a称为数列{a<sub>n</sub>}的极限，并记作
    <span class="oneline">
        <code>
            ["limt","n→∞",["rightBottom","a","n"]]
        </code>
    </span>
    读作“当n趋于无穷大时，{a<sub>n</sub>}的极限等于a”。
</p>
<h2>
    四则运算法则
</h2>
<p>
    若{a<sub>n</sub>}与{b<sub>n</sub>}为收敛数列，则{a<sub>n</sub>+b<sub>n</sub>},{a<sub>n</sub>-b<sub>n</sub>},{a<sub>n</sub>·b<sub>n</sub>}也是收敛数列，且有
    <span class="oneline">
        <code>["limt","n→∞",["join","(",["rightBottom","a","n"],"±",["rightBottom","b","n"],")"]]</code>=
        <code>
            ["limt","n→∞",["rightBottom","a","n"]]
        </code>±
        <code>
            ["limt","n→∞",["rightBottom","b","n"]]
        </code>
    </span>
    <span class="oneline">
        <code>["limt","n→∞",["join","(",["rightBottom","a","n"],"·",["rightBottom","b","n"],")"]]</code>=
        <code>
            ["limt","n→∞",["rightBottom","a","n"]]
        </code>·
        <code>
            ["limt","n→∞",["rightBottom","b","n"]]
        </code>
    </span>
    特别当b<sub>n</sub>为常数c时有
    <span class="oneline">
        <code>["limt","n→∞",["join","(",["rightBottom","a","n"],"+ c )"]]</code>=
        <code>
            ["limt","n→∞",["rightBottom","a","n"]]
        </code>+ c
    </span>
    <span class="oneline">
        <code>
            ["limt","n→∞",["rightBottom","c a","n"]]
        </code>= c
        <code>
            ["limt","n→∞",["rightBottom","a","n"]]
        </code>
    </span>
    若再假设b<sub>n≠0</sub>及
    <code>
        ["limt","n→∞",["rightBottom","b","n"]]
    </code>
    ≠0，则
    <code>
        ["bracket",["division",["rightBottom","a","n"],["rightBottom","b","n"]],"big"]
    </code>
    也是收敛数列，且有
    <span class="oneline">
        <code>
            ["limt","n→∞",["division",["rightBottom","a","n"],["rightBottom","b","n"]]]
        </code> =
        <code>
            ["division", ["limt","n→∞",["rightBottom","a","n"]], ["limt","n→∞",["rightBottom","b","n"]]]
        </code>
    </span>
</p>
<h2>
    重要性质
</h2>
<p>
    <span class="title">
        定理（唯一性）
    </span>
    若数列{a<sub>n</sub>}收敛，则它只有一个极限。
</p>
<p>
    <span class="title">
        定理（有界性）
    </span>
    若数列{a<sub>n</sub>}收敛，则{a<sub>n</sub>}为有界数列，即存在正数M，使得对一切正整数n有|a<sub>n</sub>| ≤ M。
</p>
<p>
    <span class="title">
        定理（保号性）
    </span>
    若 <code>["limt","n→∞",["rightBottom","a","n"]]</code> =
    a＞0(或＜0)，则对任何a'∈(0,a)(或a'∈(a,0))，存在正数N，使得当n＞N时有a<sub>n</sub>＞a'(或a<sub>n</sub>＜a')。
</p>
<p>
    <span class="title">
        定理（保不等式性）
    </span>
    设{a<sub>n</sub>}与{b<sub>n</sub>}均为收敛数列。若存在正数N<sub>0</sub>，使得当n＞N<sub>0</sub>时有a<sub>n</sub>≤b<sub>n</sub>，则
    <code>["limt","n→∞",["rightBottom","a","n"]]</code> ≤ <code>["limt","n→∞",["rightBottom","b","n"]]</code>。
</p>
<p>
    <span class="title">
        定理（迫敛性）
    </span>
    设收敛数列{a<sub>n</sub>},{b<sub>n</sub>}都以a为极限，数列{c<sub>n</sub>}满足：存在正数N<sub>0</sub>，当n＞N<sub>0</sub>时有
    <span class="oneline">
        a<sub>n</sub>≤c<sub>n</sub>≤b<sub>n</sub>
    </span>
    则数列{c<sub>n</sub>}收敛，且 <code>["limt","n→∞",["rightBottom","c","n"]]</code> = a。
</p>